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U. Glaeser

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Impulse Response<br />

0.2<br />

0.15<br />

0.1<br />

0.05<br />

0<br />

0 10 20 30 40 50<br />

Index (n)<br />

FIGURE 26.7 Digital filter design via Kaiser windowing: (a) shows the impulse response of an ideal lowpass filter<br />

(circles with dotted lines) and the filter designed by windowing (filled circles with solid lines). The windowed filter<br />

is the product of the ideal impulse response and the Kaiser window with β = 5 (dashed line); (b) shows the log<br />

magnitude of four filters designed using the Kaiser window with different parameters β.<br />

Frequency Sampling<br />

Another common, but naive, approach to FIR design is the method of frequency sampling. In this case,<br />

the ideal frequency response is sampled over the range −π < ω ≤ π at M + 1 points and then the inverse<br />

FFT is computed to get the order-M impulse response, which then contains the coefficients of the FIR<br />

filter. It is possible to let a few of the frequency samples be free parameters for a linear program that will<br />

optimize the resultant H(ω). This, in turn, improves the filter characteristics by making the error smaller<br />

near the cutoff frequency.<br />

Weighted Least-Squares<br />

Although frequency sampling filters and windowing designs have pretty good responses, neither one is an<br />

optimal filter. In the general optimization approach, the transition band of the frequency response should<br />

be treated as a “don’t care” region. For common frequency selective filters, the optimal filter will have a<br />

smooth behavior in the transition band even though no optimization is done in that “don’t care” region.<br />

The FIR filter can be designed by minimizing any norm with a guaranteed unique solution. The design<br />

can be generalized further by using a weighting function on the error. For example, the weight can be<br />

used in clever ways to control the error. Here is the weight definition for an inverse filter (or equalizer).<br />

The weighted design problem usually involves optimizing the norm of the error over the entire frequency<br />

domain, but that is done numerically by working on a dense frequency grid.<br />

The easiest optimization problem is the least-squares norm minimization because the partial derivatives<br />

(which are the elements of the gradient) of the least-squares norm with respect to the filter coefficients<br />

are all linear combinations of the filter coefficients. This property implies that the optimal filter can be<br />

found by solving the set of linear equations obtained by setting all those partial derivatives to zero. The<br />

solution for the weighted least-squares FIR filter is<br />

© 2001 by CRC Press LLC<br />

Windowed Impulse Response<br />

Kaiser(5.0) Window<br />

(scaled by 2*f c )<br />

Log Magnitude (dB)<br />

LPFs with cutoff at ω = 0.1π<br />

0 0.1π 0.2π 0.3π<br />

Frequency ( ω)<br />

(a) (b)<br />

I Eq(ω)<br />

0<br />

−30<br />

−60<br />

−90<br />

Kaiser(0.0)<br />

Kaiser(3.0)<br />

Kaiser(5.0)<br />

Kaiser(8.0)<br />

D(ω)<br />

= ----------------- , W<br />

HSys(ω )<br />

Eq(ω) = HSys(ω) W(ω)<br />

∂<br />

------------- W( I– H)<br />

∂b[ n]<br />

2 ∫ dω<br />

= 0,<br />

ω<br />

for n = 0, 1,…,M<br />

W 2 Ie jωn<br />

∫<br />

⎛ jω( n−k )<br />

– ∑b[<br />

k]e<br />

⎞ dω<br />

=<br />

⎝ ⎠<br />

0, for n =<br />

0, 1,…,M<br />

ω<br />

k

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